Adapting a genetic algorithm to schedule variants of the manufacturing shop.

P. R. Drake* and I. A. ChoudhryP. R. Drake* and I. A. Choudhry

Intelligent Systems Laboratory,Intelligent Systems Laboratory,

Systems Division, School of Engineering, University of Wales Cardiff,Systems Division, School of Engineering, University of Wales Cardiff,

Queen's Buildings, Cardiff, CF2 3TE.Queen's Buildings, Cardiff, CF2 3TE.

Abstract

Whilst it is well documented how genetic algorithms (GA's) can be

applied to scheduling jobs for simple manufacturing shop models,

this paper shows how they can be used to schedule jobs for some of

the more complex standard models. The simplicity of the methods

presented, and the use of proprietary spreadsheet software and GA

'add-in' to implement them, support the conjecture that GA's can

provide a highly flexible and user-friendly solution to the general

shop scheduling problem.

1 Introduction

Since the late 1980s there has been a growing interest in genetic

algorithms (GA's) - optimisation algorithms based on the principles of

natural (Darwinian) evolution. As they are based on evolutionary

learning they come under the heading of Artificial Intelligence. They

have been used widely for parameter optimisation, classification and

learning in a wide range of applications. More recently, production

scheduling has emerged as an application. In particular, such

applications have started to appear in this journal (Jain and

Elmaraghy 1997) (Lee, Piramuthu and Tsai 1997). Whilst it is well

documented how GA's can be applied to simple job sequencing

problems, this paper shows how they can be implemented to

sequence jobs/operations for manufacturing shops with precedence

constraints among manufacturing tasks, alternative routeings

(machines), sequence dependent set-up times and scheduled

machine downtime. The methods presented are very simple and

easy to implement and therefore support the conjecture that GA's

can provide a highly flexible and user-friendly solution to the general

shop scheduling problem.

The work presented here is carried out using standard spreadsheet

software and an 'add-in' to provide the basic GA. The use of this

proprietary software demonstrates how simple it is to implement the

GA approach to schedule optimisation - very little software skill is

required. Spreadsheets are used widely in manufacturing industry

2

and they are particularly suitable for production scheduling because

they store and manipulate data in tables.

2 The Sequence Optimisation GA

Detailed introductions to GA's are given by Goldberg (1989) and

Davis (1991). The flowchart for the particular GA used here is given

in Figure 1. First, an initial population of randomly generated

sequences of the tasks in the schedule is created. These individual

schedules form 'chromosomes' which are to be subjected to a form of

Darwinian evolution. The size of the population is user-defined and

the fitness of each individual schedule in the population is calculated

according to a user-defined fitness function such as: total makespan;

mean tardiness; maximum tardiness; number of tardy jobs. The

schedules are then ranked according to the value of their fitness

function.

Once an initial population has been formed, 'selection', 'crossover'

and 'mutation' operations are repeatedly performed until the fittest

member of the evolving population converges to an optimal fitness

value. Alternatively, the GA may run for a user defined number of

iterations.

3

The 'selection' process consists of selecting a pair of parents from the

current population using a rank-based mechanism to control the

probability of selection. These parents then mate to produce a child

by applying (in this particular implementation) the 'uniform order-

based' crossover operator (Davis 1991). This is based on using a

randomly generated bit string (zero-one) template to determine for

each parent which genes are carried forward into their child. The

operator is explained by the example in Figure 2. Here, the bit string

template contains a one at positions {2, 3, 5, 6}. The genes (jobs) in

these positions in parent-1 are carried straight across into the child.

The template contains zeros at positions {1, 4, 7, 8}. In these

positions parent-1 has genes {1, 4, 7, 8}. These genes are copied

into the child in the same order as they appear in parent-2 {8, 4, 7,

1}. The proportion of the jobs in the child coming from the first

parent is defined by the crossover rate which has a value between 0

and 1.

Following crossover, the resultant child may be mutated. The

particular mutation operation used here is 'order-based' mutation. In

this operation two jobs change positions in the chromosome. The

probability that a job is mutated is defined by the user defined

mutation rate which lies in the range 0 to 1. The purpose of mutation

is to ensure that diversity is maintained within the population. It

gives random movement about the search space thus preventing the

GA becoming trapped in 'blind corners' or 'local optima'. Finally, the

new 'child' schedule replaces the weakest (least fit) schedule in the

current pool or 'population' of schedules.

4

A population size of 50, a crossover rate of 0.65 and a mutation rate

of 0.006 are used in the experiments presented here.

3 Why use GA's for Shop Scheduling ?

It is a characteristic of the schedule optimisation process that once

fairly good solutions have been formed their features will be carried

forward into better solutions, and lead ultimately to optimal

solutions. It is in the nature of the GA that new solutions are formed

from the features of known good solutions. Therefore, it follows that

GA's are particularly attractive for scheduling.

Compared with other optimisation methods, GA's are suitable for

traversing large search spaces since they can do this relatively

rapidly and because the mutation operator diverts the method away

from local minima, which will tend to become more common as the

search space increases in size. Being suitable for large search spaces

is a useful advantage when dealing with schedules of increasing size

since the solution space will grow very rapidly, especially when this is

compounded by such features as alternative machines/routes. It is

important that these large search spaces are traversed as rapidly as

possible to enable the practical and useful implementation of

automated schedule optimisation. If the optimisation is done quickly

then production managers can try out 'what-if' scenarios and detailed

sensitivity analysis as well as being able to react to 'crises' as soon

as possible.

5

For a simple n jobs and m machines schedule the upper bound on the

number of solutions is (n!)m. Without n and m being very large this

value soon becomes massive, e.g. for 20 jobs and 5 machines the

value is 8.52x1091. Traditional approaches to schedule optimisation

such as mathematical programming and 'branch and bound' are

computationally very slow in such a massive search space.

4 Example 1: A Flow Shop

Table 1 gives the job data for this example and the objective is to

minimise the makespan for the schedule. When the GA is applied to

this example the best sequence is 3-6-4-7-2-8-1-5 which yields a

makespan of 584 units of time. The result of applying the 'network

scheduling technique' to this example was recently reported and the

best sequence found was 3-6-4-7-8-2-1-5 which produces a

makespan of 595 units of time (Akpan 1996). This was also the best

result reported when using a heuristic method (Campbell, Smith and

Dudek 1970).

5 Precedence Constraints

The normal sequence optimising GA will rearrange freely the jobs in a

schedule. However, in most production scheduling situations there

will be precedence constraints on the order of operations within a

particular job. For example, holes must be drilled before they are

tapped. To implement such constraints, a precedence table is

6

introduced as in Table 2 for the example considered in the next

section. This table shows that, for example, task 5 must be done

before tasks 6 and 7. When the GA creates a child the precedence

table is checked. If any precedence constraints are broken then tasks

which must be performed earlier in the sequence are moved up the

schedule.

6 Example 2: Alternative Machines

There may be a choice of machines that can be used for a particular

operation as illustrated in Table 3. In this example operation 2 for job

A can be performed on machine 1 or 3, operation 2 for job B on

machine 1 or 2, and so on. Table 4 presents the same information in

the form of a 20 job-operation sequence. Note, five of these

operations {tasks 3,7,13,15,19} will not in fact be performed. They

are present in the table solely for the purpose of making the

alternatives available to future generations. The first occurrence of a

duplicated operation is taken to be an 'actual' occurrence that must

be included in the calculation of the fitness function while subsequent

occurrences are ignored. However, all occurrences are included in

the crossover and mutation operations.

Using integer programming, Jiang and Hsiao (1994) produce two

optimised schedules for this example - one to minimise the total

absolute deviation from the due dates and one to minimise the total

finishing times of the individual jobs. They show that the minimum

total absolute deviation is four time units and the minimum total

7

finishing time is ten time units. The GA produced the same results.

Figures 3 and 4 show the schedules generated to optimise the two

fitness functions. Table 5 illustrates how the GA presents the optimal

solution for the minimum total finishing times schedule. As explained

before, this is a sequence of 20 operations including 5 duplicated

operations that are to be ignored.

7. Example 3: Sequence Dependent Set-up Times

Inter-operation set-up times may depend on the type of operation

that has just been completed, as well as on the type of operation to

be performed. In such situations it is not valid to absorb the set-up

time for a job into its processing time. To accommodate sequence

dependent set-up times, look-up tables containing the times are

introduced. Then, fitness functions such as total makespan (lead-

time) can be calculated for a schedule by combining process times

with the appropriate set-up times in the look-up table.

Table 6 presents an example schedule with the corresponding

sequence dependent set-up times being given in Tables 7 to 9. The

smallest total makespan found by the GA is 45 time units with the

task sequence {1,7,8,9,4,5,10,11,12,2,3,6}. Again, it is a relatively

simple matter to implement the fitness function required to enable

the GA to optimise this schedule.

8. Example 4: Scheduled Machine Downtime

8

Scheduled machine downtime may be requested for preventive or

reactive maintenance. In the model considered here, the downtime

must be inserted into the production schedule according to the

following two rules.

i. If there is no production scheduled for a machine at the start of

a requested downtime, then the downtime is allocated as

requested. Any production scheduled to start during the

downtime is moved to the end of the downtime.

ii. If production is already scheduled at the start time of the

requested downtime, then the downtime is moved to the end

of this operation.

Other rules may be used as desired. The important principle being

demonstrated here is how these rules are applied with the GA.

Consider, the 3-job, 4-machine job shop example in Table 10 taken

from Pinedo (1995). The precedence constraints for this example are

defined in Table 11. The best makespan found by Pinedo for this

example is 28. He uses disjunctive graphs and proves that this

solution is optimal. The GA finds the same solution but with a

different sequence of operations. The Gantt charts for the two

solutions are given in Figure 5.

Now consider that before the execution of the schedule the

maintenance department request machines 1, 2 and 4 for the times

9

given in Table 12. This machine downtime is inserted into the

schedule according to the above two rules to give the schedule in

Figure 6. This increases the makespan to 55 time units. Rerunning

the GA to optimise this schedule, the makespan is reduced to 51 as

shown in Figure 7.

9 Conclusion

The adaptation of a GA to schedule jobs in manufacturing shops with

precedence constraints among tasks, alternative routeings, sequence

dependent set-up times and scheduled machine downtime has been

demonstrated. The simplicity of the methods used supports the

conjecture that GA's can provide a highly flexible and user-friendly

solution to the general shop scheduling problem. The use of

proprietary spreadsheet software and a proprietary 'add-in' to

implement the GA has shown how simple it is to implement this

approach to scheduling.

References

AKPAN, E. O. P., 1996, Job-shop sequencing problems via network

scheduling technique, International Journal of Operations and

Production Management, 16, 76-86.

10

CAMPBELL, H. G., SMITH, M. L., DUDEK, R. A., 1970, A heuristic

algorithm for the N jobs, machine sequencing problem, Management

Science, 16, B630-B637.

DAVIS L, 1991, Handbook of Genetic Algorithms, New York: Van

Nostrand Reinhold.

GOLDBERG, D. E., 1989, Genetic Algorithms in Search, Optimisation

and Machine Learning, Reading, MA: Addison-Wesley.

JAIN A. K., ELMARAGHY H.A., 1997, Production

scheduling/rescheduling in flexible manufacturing, International

Journal of Production Research, 35, 281-309.

JIANG J. and HSIAO W., 1994, Mathematical Programming for the

Scheduling Problem with Alternative Process Plans in FMS, Computers

& Industrial. Engineering, 27, 15-18.

LEE C.-Y., PIRAMUTHU S. and TSAI Y. -K., 1997, Job shop scheduling

with a genetic algorithm and machine learning, International Journal

of Production Research, 35, 1171-1191.

PINEDO, M., 1995, Scheduling: Theory, Algorithms and Systems,

Prentice Hall, New Jersey.

Acknowledgements

11

The authors would like to thank the Pakistan Government for their

assistance in this research.

12

Figure 1: The Genetic Algorithm (GA) for Schedule Optimisation

13

Position in Chromosome1 2 3 4 5 6 7 8

Parent-1

Binary Template

Parent-2

Child

1 2 3 4 5 6 7 8

0 1 1 0 1 1 0 0

8 6 4 2 7 5 3 1

8 2 3 4 5 6 7 1

Figure 2: Schedule Chromosomes and the Crossover Operator

Figure 3 : Schedule to minimise absolute deviation from due dates.

14

Figure 4 : Schedule to minimise total finishing times.

Figure 5 : Gantt Chart for Pinedo’s Solution and the GA Solution

15

Figure 6 : Modified schedule with machine downtime inserted

Figure 7 : Modified schedule after rerunning the GA

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Table 1: Job Data for 8-job, 7-machine flow line example.

Job List

Processing Times for Machine

1 2 3 4 5 6 71 13 79 23 71 60 27 22 31 13 14 94 60 61 573 17 1 - 23 36 8 864 19 28 10 4 58 73 405 94 75 - 58 - 68 466 8 24 3 32 4 94 897 10 57 13 1 92 75 298 80 17 38 40 66 25 88

Table 2: Precedence Constraints for Tasks

Task Numb

er

Must follow

12 13 14 2 35 6 57 5 8 6 79 10 911 10121314 12 1315 12 1316 14 151718 1719 1720 18 19

17

Table 3: The Alternative Machines Schedule

Job-Operati

on

Processing time

for machine

DueDate

1 2 3 4A-1 2A-2 1 2A-3 1 6B-1 1B-2 1 2B-3 1 3C-1 2C-2 2C-3 1 6D-1 1 2D-2 2 1D-3 1 6E-1 1E-2 2 1E-3 2 6

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Table 4: GA Representation of the Example Alternative Machines Schedule

Job Numb

er

Operation

Machine Process Time

Contribution to

Fitness Function

Task Number

*

A 1 1 2 2 1A 2 1 1 1 2A 2 3 2 - 3A 3 4 1 1 4B 1 2 1 1 5B 2 1 1 1 6B 2 2 2 - 7B 1 1 1 1 8C 1 3 2 2 9C 2 4 2 2 10C 3 1 1 1 11D 1 3 1 1 12D 1 4 2 - 13D 2 2 2 2 14D 2 3 1 - 15D 3 2 1 1 16E 1 2 1 1 17E 2 2 2 2 18E 2 4 1 - 19E 3 3 2 2 20

* This column represents the chromosome used in the crossover and mutation operations.

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Table 5: GA Optimised Schedule to Minimise Total Finishing Time

Job Numb

er

Operation

Machine Process Time

Contribution to

Fitness Function

Task Number

*

D 1 3 1 1 12A 1 1 2 2 1A 2 1 1 1 2D 1 4 2 - 13A 2 3 2 - 3D 2 3 1 1 14E 1 2 1 1 17E 2 4 1 1 19B 1 2 1 1 5B 2 1 1 1 6D 2 3 1 - 14D 3 2 1 1 16E 2 2 2 - 18E 3 3 2 2 20B 2 2 2 - 7A 3 4 1 1 4C 1 3 2 2 9C 2 4 2 2 10B 3 1 1 1 8C 3 1 1 1 11

* This column represents the chromosome used in the crossover and mutation operations.

20

Table 6: Schedule Data for the Sequence Dependent Set-up Times Example.

Job Operation

Machine Part Type

Process Time

Task Number

*1 1 B 1 4 11 2 A 1 7 21 3 C 1 6 32 1 B 2 9 42 2 C 2 5 52 3 A 2 4 63 1 A 3 1 73 2 B 3 2 83 3 C 3 10 94 1 A 4 8 104 2 C 4 4 114 3 B 4 5 12

* This column represents the chromosome used in the crossover and mutation operations.

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Table 7: Sequence Dependent Set-up Times for Machine 1

To Part TypeFrom Part

Type 1 2 3 4

None 2 3 2 41 0 3 1 52 1 0 3 23 2 4 0 14 2 4 2 0

Table 8: Sequence Dependent Set-up Times for Machine 2

To Part TypeFrom Part

Type 1 2 3 4

None 2 1 3 11 0 2 3 52 5 0 1 33 4 2 0 14 3 3 4 0

Table 9: Sequence Dependent Set-up Times for Machine 3

To Part TypeFrom Part

Type 1 2 3 4

None 1 3 2 21 0 3 2 52 3 0 4 33 2 2 0 44 2 5 3 0

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Table 10: Processing times and machine sequences for

3-job, 4-machine example

Operation Processing Times

Machine Sequence

Job 1st 2nd 3rd 4th 1st 2nd 3rd 4th1 10 8 4 - 1 2 3 -2 8 3 5 6 2 1 4 33 4 7 3 - 1 2 4 -

Table 11: Precedence Constraint Table

Job-operation

Must follow job-

operation1-11-2 1-11-3 1-22-12-2 2-12-3 2-22-4 2-33-13-2 3-13-3 3-2

Table 12 : Requested Machine Downtimes

Requested DowntimesMachin

eStart Finish Durati

on

23

1 10 20 102 15 30 154 10 40 30

24